Introductory description

The theory of partial differential equations (PDE) is important both in pure and applied mathematics. On the one hand they are used to mathematically formulate many phenomena from the natural sciences (electromagnetism, Maxwell's equations) or social sciences (financial markets, Black-Scholes model). On the other hand since the pioneering work on surfaces and manifolds by Gauss and Riemann partial differential equations have been at the centre of many important developments on other areas of mathematics (geometry, Poincare-conjecture).

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The course will start with an introduction to PDES, outlining what PDEs actually are, how some of them can be classified, how do they arise in physics and the life sciences, and what the notion of well-posedness means. It will mainly focus on three classical equations – the heat equation, the wave equation and the Laplace equation – on unbounded and bounded domains.

Learning outcomes

By the end of the module, students should be able to:

• understand and apply fundamental physical principles to derive certain types of PDEs.
• the ability to discriminate between various types of PDEs and comprehend the distinctions in their inherent characteristics.
• study solutions to the three key types of PDEs on unbounded and bounded domains
• deep understanding on how the properties of solutions to PDEs sensitively depend on the type.

Indicative reading list

A script based on the lecturer's notes will be provided. For further reading you may find the following books useful (sections of relevance will be pointed out in the script or in the lectures):
S Salsa: Partial differential equations in action, from modelling to theory. Springer (2008).
A Tveito and R Winther: Introduction to partial differential equations, a computational approach. Springer TAM 29 (2005).
W Strauss: Partial differential equations, an introduction. John Wiley (1992).
JD Logan: Applied partial differential equations. 2nd edt. Springer (2004).
MP Coleman: An introduction to partial differential equations with MATLAB. Chapman and Hall (2005).
M Renardy and RC Rogers: An introduction to partial differential equations, Springer TAM 13 (2004).
LC Evans: Partial differential equations. 2nd edt. American Mathematical Society GMS 19 (2010).

Subject specific skills

At the end, students will be familiar with the notion of well-posed PDE problems and have an idea what kind of initial or boundary conditions may be imposed for this purpose. Students will have studied some techniques which enable you to solve some simple PDE problems. They will also understand that properties of solutions to PDEs sensitively depend on the type.

Transferable skills

The module provides technical competence in solving basic partial differential equations that feature at least as building blocks in applications. There are aspects of critical thinking and creativity related to analysing and solving PDE problems.